Preheat the oven to 400 degrees. Clean beets and drizzle with olive oil over foil. Place on a baking sheet and bake for 35-45 minutes. Remove from oven.

Submerge eggs into boiling water for 12 minutes and then immediately place in ice water for 15 minutes. Remove one at a time and peel. Place in a medium bowl or large glass jar.

Once the beets have cooled for at least 10 minutes, peel and cut into dice-sized cubes. Place beets in a medium saucepan with white vinegar, apple cider vinegar, sugar, water, and a pinch of salt. Heat over medium/high until the mixture starts to steam. 

Pour mixture over the eggs, cover, and place in the fridge for 24 hours and up to 48. Every once in a while move the bowl a bit to re-position the eggs. 

Slice the eggs in half lengthwise and reserve the yolks in a small bowl. Add mayonnaise and sweet, hot mustard to the yolks and mix until smooth. Use an immersion blender if too lumpy. Place yolks in a piping bag and pipe into the eggs until it overflows. 

Slice the shallots thinly and place in a small frying pan over medium heat with butter and sugar stirring every half minute or so. Once the shallots have browned, add in the sherry, stir together and remove from heat. Place a bit of caramelized shallot and black sesame seeds on the top of each egg.